This does seem as though the WIFI bandwidth may be insufficient. I believe the ability of the WIFI to handle multiple users is dependent on the overall speed network DSL service. If the network is shared throughout the school, activities over the entire network may affect the classroom. Students streaming music or video (either in the class or in the cafeteria or library for instance) might affect the 15 students.
Have you tried the emulator option? The emulator generally works extremely well, however both it and the USB option require ports 8001 and 8004 be available for App Inventor...they might be blocked by your IT department and might be why you are unable to work with USB. Much depends on the restrictions your IT department places on the Firewall.
Regarding USB, have you downloaded the required USB drivers to the computers and then set the devices themselves to allow USB Debugging (Set in the individual devices using the Android Settings of the device)?
Here is some documentation that your IT might find of value
Any of this stuff giving you and your students relief Doug?
(please keep the conversation on the forum ... i.e. no emails)
Regards,
Steve